Theorem necon3abii 2547

Description: Deduction from equality to inequality. (Contributed by NM, 9-Nov-2007.)

Hypothesis

Ref	Expression
necon3abii.1	⊢ (A = B ↔ φ)

Assertion

Ref	Expression
necon3abii	⊢ (A ≠ B ↔ ¬ φ)

Proof of Theorem necon3abii

Step	Hyp	Ref	Expression
1		df-ne 2519	. 2 ⊢ (A ≠ B ↔ ¬ A = B)
2		necon3abii.1	. 2 ⊢ (A = B ↔ φ)
3	1, 2	xchbinx 301	1 ⊢ (A ≠ B ↔ ¬ φ)

This proof depends on syntax axioms:  ¬ wn 3   ↔ wb 176   = wceq 1642   ≠ wne 2517

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 177  df-ne 2519

This theorem is used by:  necon3bbii  2548  necon3bii  2549  necompl  3545  n0f  3559  foconst  5281